View of THE KARUSH KUHN TUCKER OPTIMALITY CONDITIONS IN A CLASS OF GENERALIZED CONVEX OPTIMIZATION PROBLEMS WITH INTERVAL-VALUED MULTIOBJECTIVE FUNCTIONS

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Vol.03, Issue 09, Conference (IC-RASEM) Special Issue 01, September 2018 Available Online: www.ajeee.co.in/index.php/AJEEE

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THE KARUSH KUHN TUCKER OPTIMALITY CONDITIONS IN A CLASS OF GENERALIZED CONVEX OPTIMIZATION PROBLEMS WITH INTERVAL-VALUED

MULTIOBJECTIVE FUNCTIONS Deepak Singh¹, Harsha Atre²

1NITTTR, Ministry of HRD, Govt. of India, Bhopal, M.P.

2Department of Mathematics, Bhopal ,M.P

Abstract In this paper our main motive is to extend KKT optimality conditions to non convex multiobjective optimization problems with an interval-valued objective functions.

Using the order relations "≤𝐿𝑈" and "≤𝑈𝐶" the solution concepts for interval valued optimization problems is given. Further we discuss the concept of LU-preinvexity and invexity for interval-valued functions. And finally the KKT optimality condition is derived for multiobjective optimization problem with an interval-valued objective functions under the assumption of LUpreinvexity and invexity.

Keywords KKT optimality condition, Intervalvalued functions, Type-I and Type-II solution, LU and UC invexity.

1. INTRODUCTION

Theory and mathematical modeling are the major constituents for any solution of the optimization problem. To determine the coefficients of objective functions involved is usually a tedious task in practice. There are two deterministic optimization models to deal with uncertain data viz. robust optimization Ben-Tal et al. [2], and another is interval valued optimization Ben-Israel and Robers [1].

The interval-valued optimization problem is one of the type of inexact linear programming problems. Charnes et al. [3] considered the linear programming problems in which the right-hand sides of linear inequality constraints were taken as closed intervals. In a paper Stancu- Minasian and Tigan[16] obtained solutions for interval valued optimization problems. Ishibuchi and Tanaka [7]

proposed the ordering relation between two closed intervals by considering the maximization and minimization problems separately. Mraz[5] proved algorithms to compute the exact upper bound and lower bound for linear programming problems with interval coefficients. Chanas and Kuchta [21]presented an approach to unify the solution methods proposed in Ishibuchi and Tanaka [7] and in Rommelfanger and Hanuscheck[8].

Oliveria and Antunes[4] provided an overview of multiobjective linear programming problems with interval coefficients by illustrating many numerical examples. Lai and Huang etal.

[25] proposed an interval parameter fuzzy

nonlinear optimization model forstream water quality management under uncertainty.

The Karush-Kuhn-Tucker optimality conditions play an important role in area of optimization theory and have been studied for over a century. For interval valued optimization problems, the KKT optimality conditions are also studied in many recent publications. Recently Wu ([13], [14]) have studied KKT-optimality conditions for interval valued optimization problems. Also Chalco-Cano et al.[24]

studied the KKT-optimality conditions of interval valued optimization problem via generalized derivative.

In [14] Wu proposed the concept of non dominated solution in vector optimization problems, Wu [14] has proposed a solution concept for optimization problems with an interval- valued objective function based on a partial ordering on the set of all closed intervals and then the interval-valued Wolfe duality theory [12] and Lagrangian duality theory [11] for interval-valued optimization problems. He has established the KKT conditions [10] for an optimization problem with an interval- valued objective function under the assumption of LUconvexity. Recently, Wu [9] has studied the duality theory for interval-valued linear programming problems.

Zhang [17] recently extended the KKT optimality conditions to nonconvex optimization problems with an interval- valued objective function and he extended the concepts of preinvexity and invexity for a real-valued objective functions,

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2 Hanson [15] to an interval-valued function and presented the concepts of LU-preinvexity and invexity for an interval-valued function.

In this paper our main motive is to study KKT optimality conditions to non convex optimization problems with an interval-valued multiobjective function.

The remaining paper is sectioned as:

In Section 2, we introduce some preliminaries of interval arithmetic and the concept of weak differentiability for intervals valued functions. Moreover by using the concept of order relations "≤𝐿𝑈"

and "≤𝑈𝐶" the solution concepts for interval valued optimization problems are given. Also the concept of LU-convexity and UC-convexity are provided. In Section 3, we introduce the concept of LU- preinvexity and invexity for an interval- valued multi objective function.

In Section 4, the KKT optimality conditions are derived for an optimization problem with an intervalvalued multiobjective function under the assumption of LU-preinvexity and invexity.

2. PRELIMINARIES

2.1 Some concepts of the interval- valued functions

Let us denote by 𝒯 the class of all closed intervals in 𝑅. If 𝐴 is a closed interval, we also adopt the notation where 𝑎 𝐿 and 𝑎 𝑈means the lower and upper bound of 𝐴, respectively

and Then, by definition, we have

I.

II.

Therefore, we also see that

We also see that𝑘𝐴 = {𝑘𝑎: 𝑎 ∈ 𝐴} =

Wherek is a real number. For more details on the topic of interval analysis, we refer to Moore [18, 19] and Alefeld and Herzberger [6].

Definition 2.1 Let 𝐴 = be two closed intervals in 𝑅. We say that 𝐴 is less than or equal to B and write 𝐴 ≤ 𝐵if and only if 𝑎 𝐿 ≤ 𝑏 𝐿 and𝑎 𝑈 ≤ 𝑏 𝑈.

Also we say that 𝐴is less than 𝐵 and write 𝐴 < 𝐵if and only if 𝐴 ≤ 𝐵and𝐴 ≠ 𝐵.

Equivalently, 𝐴 < 𝐵 if and only if

The function called an

interval-valued function, i.e., is a closed interval in 𝑅 for each The interval-valued function 𝑓 can also be written as

]where are real-valued functions defined on and

satisfy for each

Definition 2.2[13] Let 𝑋 be a nonempty convex subset of 𝑅 𝑛 and 𝑓 be an interval- valued function defined on𝑋. We say that 𝑓is convex at 𝑥 ∗ if

For each 𝑥 ∈ 𝑋 and 𝜆 ∈ (0,1) Solution Concept

Consider an interval-valued optimization problem (IVOP) as

are interval valued functions and the relation "<" ," ≤ "

defined in definition 2.1.According to Definition is a feasible solution of problem and

us denote by X the set of feasible solution of problem (IVOP).

Similar to usual multi problems, often there does not exist a point 𝑥 ∈ 𝑋 to minimize all of the objective functions, simultaneously. Therefore we need to define another notion of optimal solution, named Pareto optimal (efficient) solution.

Now consider the multiobjective programming problem with interval- valued functions

Subject to Where each

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3 and

Definition 2.3 [13] Let𝑥 ∗ be a feasible solution, i.e. 𝑥 ∗ ∈ 𝑋 . We say that 𝑥 ∗ is a Type-I solution of problem (MVIP) if there exists no 𝑥̅ ∈ 𝑋 such that

Next, we introduce solution concept following from Ishibuchi and Tanaka [7].

Let be two closed intervals in 𝑅, the center of 𝐴 is and the half-width of 𝐴 is We can also use to denote 𝐴.Ishibuchi and Tanaka [7] have proposed the ordering relation between the two closed intervals of and

Definition 3.1[21, 22] A set is to said be invex if there exists a vector functio such that

Definition 3.2[22, 23 be an invex set with respect to We say that

𝑓 is preinvex if

Definition 3.3[15] be an invex

set with respect to

We say that 𝑓 is invex if

For convenience, set of all interval-valued function defined on Y is denoted by (𝑌) , an invex set with respect to the function 𝜂: 𝑌 × 𝑌 → 𝑅 𝑛 is called an 𝜂- invex set and an invex function with respect to the function𝜂: 𝑌 × 𝑌 → 𝑅 𝑛 is called an 𝜂-invex function.

Remark 3.1 Pini [18] has shown that, if 𝑓a realvalued function defined on an invex set 𝑌and if it is preinvex and differentiable, then 𝑓is also an η-invex function, but the converse is not true in general. The relationships between invex and convex sets,

Definition 2.4 [13] Let𝑥 ∗ be a feasible solution, i.e. 𝑥 ∗ ∈ 𝑋. We say that 𝑥 ∗ is a Type-II solution of problem (MIVP) if there exists no 𝑥̅ ∈ 𝑋 such that

3. PREINVEXITY AND INVEXITY OF INTERVAL-VALUED FUNCTIONS

The concept of convexity is an important feature of the optimization theory. In recent years, the convexity concept has been extended in many areas. An important extension of a convex function is the introduction of a preinvex function, which was introduced by Weir and Mond [22] and by Weir and Jeyakumar [23].

Yang [26] has established the characterization of prequasi-invex functions under the conditions of lower semicontinuity, upper semi continuity, and semi strict prequasi-invex respectively.

Preinvex and convex functions, and invex and convex functions are as follows.

(1) If 𝑌 is a convex set, then it is also an invex set with 𝜂(𝑥, 𝑦) = 𝑥 − 𝑦 for all 𝑥, 𝑦 ∈ 𝑌, but the converse is not true.

(2) Let 𝑓 ∈ 𝐼 (𝑌) be a convex function, then 𝑓is also a preinvex function with 𝜂(𝑥, 𝑦) = 𝑥 − 𝑦 for all 𝑥, 𝑦 ∈ 𝑌, but the converses are not true.

However, Weirand Jeya kumar [22]

have proved that if a preinvex function 𝑓 has a unique global minimizer at𝑥 ∗ ∈ 𝑌, then 𝑓is also convex at 𝑥 ∗ .

Now, we extend the concepts of Preinvexity and invexity to interval-valued vector functions.

Definition 3.4 be an 𝜂-invex set, and 𝑓 ∈ 𝐼(𝑌) be an interval-valued functions. Then 𝑓is LUpreinvex at 𝑥 ∗ with respect to 𝜂 if

Similarly, we can define the UC- preinvexity.

Definition 3.5 be an 𝜂-invex set, and 𝑓 ∈ 𝐼(𝑌) be an interval-valued functions. Then 𝑓is UCpreinvex at 𝑥 ∗ with respect to𝜂 if

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4 Proposition 3.2 [17] Letbe an 𝜂-invex subset of 𝑅 𝑛 and𝑓 ∈ 𝐼(𝑌) be an interval- valued functions. Then we have the following properties with respect to the same

1. 𝑓 is LU – preinvex at𝑥 ∗ if and only if 𝑓 𝐿 and 𝑓 𝑈 are preinvex at 𝑥 ∗ . 2. 𝑓 is UC – preinvex at𝑥 ∗ if and only

if 𝑓 𝑈 and 𝑓 𝐶 are preinvex at 𝑥 ∗ . Proposition 3.3 be an 𝜂-invex set, and the interval-valued functions𝑓(𝑥)

= be an LU-preinvex

function. If 𝑥 ∗ ∈ 𝑋is the unique type-I minimize of f [23], then𝑓is LU-convex at𝑥 ∗ Proof Since 𝑓𝑘 is LU-preinvex, there exists 𝜂 such that for any 𝑥, 𝑦 ∈ 𝑌, 𝜆 ∈ [0,1].

Particularly when we have Since 𝑥 ∗ is the unique type-I minimize by Remark 3.1(2), there is no 𝑥 ≠ 𝑥 ∗ such that

Hence, the interval-valued multiobjective functions𝑓is LU-convex at 𝑥 ∗ .

Definition 3.6 Let be an η-invex set and the interval-valued function𝑓(𝑥) = 𝐼(𝑌). Then f is invex at 𝑥 ∗ if the real-valued functions are invex at.

Preposition 3.4 Let 𝑌 ⊆ 𝑅 𝑛 be an η-invex

set, if is a

weakly continuously differentiable and LU-preinvex intervalvalued function. Then 𝑓𝑘 is also an η-invex intervalvalued function.

Proof Since 𝑓 is an LU-preinvex function, then𝑓 𝐿 and 𝑓 𝑈are preinvex on Y by using

Preposition 3.2(i). From the assumption of weakly continuously differentiability, it can be shown that 𝑓 𝐿 and 𝑓 𝑈are continuously differentiable by Definition 2.3. By using Remark 3.1 and Definition 3.6, 𝑓isalso an ηinvex interval-valued function with respect to the same η.

4. KARUSH -KUHN-TUCKER (KKT) CONDITIONS

Consider a real-valued optimization problem (RVOP)

Theorem 4.1[13] Assume that the real- valued constrain function𝑔𝑖 , i = 1,2, … , 𝑚 of problem (IVOP) satisfies the KKT assumption at 𝑥 ∗ and the objective function 𝑓𝑘: 𝑅 𝑛 → 𝒯 is LU-preinvex and weakly continuously differentiable at𝑥 ∗ .If there exits Lagrange multipliers 0 < 𝜆 𝐿 , 𝜆 𝑈 ∈ 𝑅 and0 ≤ 𝜇𝑖 ∈ 𝑅 , 𝑖 = 1,2, … 𝑚 . Such that

Then 𝑥 ∗ is a type-I and type-II optimal solution of problem (IVOP).

Theorem 4.2 be an 𝜂-invex set, 𝑥 ∗ ∈ 𝑌, 𝑓𝑘 is LU-preinvex at 𝑥 ∗ and

for all invex function

with respect to the same

weakly continuously differentiable at and continuously differentiable function .Then the following statement hold true.

(A) Assume that the objective function preinvex at𝑥 ∗ for 𝑘 = 1,, 𝑟. If there exits Lagrange multipliers

Then𝑥 ∗ is a type-I Pareto optimal solution of problem (MVIP).

(B) Assume that the objective function 𝑓𝑘 ∈ 𝐼(𝑌) is UC-preinvex at𝑥 ∗ for 𝑘 = 1, ⋯ , 𝑟.

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5 If there exits Lagrange multipliers such that and

Then 𝑥 ∗ is a type -II Pareto optimal solution of problem (MVIP).

Proof Since𝑓𝑘 are weakly continuously differentiable at𝑥 ∗ . We define real-valued function

According to assumption,𝑓 is LU-preinvex and weakly continuously differentiable at 𝑥 ∗ . Therefore, by Proposition 3.4 the multi objective interval-valued function is also invex at 𝑥 ∗ . The real valued function is also invex at 𝑥 ∗ by Proposition3.7. Then

By using (i) and (ii) from Theorem 4.1, we have

According to Theorem 4.3 [17], is an optimal solution of the function 𝑓̌ subject to the same constrains of problem (MIVP), i.e.

Let 𝑥 ∗ is not a type-I Pareto optimal solution of problem (MIVP).

From Definition 2.3, there exits an 𝑥̌ ∈ 𝑋 such that

is satisfied. Then, from

Which contradict (7).It shows 𝑥 ∗ is a type- I Pareto optimal solution of problem (MIVP)

The result (B) can be easily obtained by considering

Then 𝑥 ∗ is a type-II Pareto optimal solution of problem (MIVP).

Theorem 4.3 be an 𝜂-invex set, 𝑥 ∗ ∈ -preinvex at 𝑥 ∗ and for all invex function with respect to the same 𝐼(𝑋)weakly continuously differentiable at 𝑥 ∗ for 𝑘 = 1,, 𝑟 and continuously differentiable function If there exits Lagrange multipliers 0 ≤ 𝜇𝑖 ∈ 𝑅 , 𝑖 = 1,2, … 𝑚. Such that

Then 𝑥 ∗ is a type-I and type-II Pareto optimal solution of problem (MIVP).

Proof Let0 < for 𝑘 = 1,2, ⋯ , 𝑟by using (i) , (ii) and (iii) ,we have

Let

By theorem 4.2, we can prove that 𝑥 ∗ is a type-I and type-II Pareto optimal solution of problem (MIVP).

Theorem 4.4 bean 𝜂 invex set,𝑥 ∗ ∈ 𝑋 , let If for every 𝑖 = 1,2, ⋯ , 𝑚, 𝜇 ∗ =

If there exits Lagrange multipliers Such that

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6 Proof The proof is similar to theorem 6.4 in [13]. Since 𝑓𝑘 is LU preinvex and weakly continuously differentiable at 𝑥 ∗ by proposition 3.2 (i) and definition 2.3 [17]. From, remark 3.1 [17], it can be shown that are invex and continuously differentiable at 𝑥 ∗ with respect to same 𝜂-invex, using this

condition as also invex

and continuously differentiable at 𝑥 ∗ . We define

Then, it can be concluded that 𝑥 ∗ is a type-I and typeII Pareto optimal solution of problem (MIVP) by theorem 4.2.

Theorem 4.5 Assume that the real-valued

constrain function of

problem (IVOP) satisfies the KKT assumption at 𝑥 ∗ and the intervalvalued objective function is LU- preinvex and continuously H- differentiable gradient of 𝑓at 𝑥 is

the partial derivative is a closed interval at 𝑥 ∗ .If there exits Lagrange multipliers 1,2, … 𝑚 . Such that

Hanson [15] has proved the following result for problem (RVOP).

Theorem 4.6 be an 𝜂-invex set, 𝑥 ∗ ∈ -preinvex at 𝑥 ∗ and for all invex function with respect to the same 𝜂 . If 𝑓𝑘continuously H-differentiable interval-valued [14]

function at 𝑥 ∗ for 𝑘 = 1, 𝑟 and continuously differentiable function .If there exits Lagrange multipliers

Such that

Then 𝑥 ∗ is a type-I and type-II optimal solution of problem(MIVP).

Proof let be an

intervalvalued function and be H- differentiable [14] at the gradient of 𝑓𝑘 at 𝑥0 is the partial

derivative is a

closed interval. From (i), we have

For all 𝑗 = 1,2, ⋯ , 𝑛 . The above equalities can be written as

Then , we have

From (8) and theorem 4.2 by taking We can prove that 𝑥 ∗ is a type –I and type-II Pareto optimal solution of problem (MIVP).

6. CONCLUSIONS

In this paper, we have introduced the concepts of LUpreinvexity and invexity for interval-valued functions. These concepts are the generalizations of preinvexity and invexity for real-valued functions. We also discussed the properties of LU-convexity, LU-preinvexity and invexity for interval- valued multi objective functions. We proved the KKT optimality conditions under the multi objective function with the assumptions of LU-preinvexity and invexity. These results extended the earlier results to interval-valued multi objective optimization problems. In future this same process can be used for equality constraint and inter-valued constant function.

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